The Indefinite Integral

Abstract

The theory of indefinite integrals is one of the basic topics of mathematical analysis. In this chapter, we study the main properties of the indefinite integral and give tables of the integrals of the main functions of analysis. We will find that sometimes the integrals of the elementary functions cannot be expressed in terms of finite combinations of the familiar algebraic and elementary transcendental functions. We shall discuss various basic techniques of integration: integration by substitution and by parts, which can often be used to transform complicated integration problems into simpler ones. Moreover, we shall discuss methods for integrating arbitrary rational functions; in particular, we will explain the method of Ostrogradsky. Finally, the evaluation of integrals involving irrational algebraic functions, quadratic polynomials, trigonometric functions, etc., will also be considered.